A370822 Lexicographically earliest sequence of positive integers such that all equal terms appear at mutually coprime indices.

1, 1, 1, 2, 1, 3, 1, 4, 2, 5, 1, 6, 1, 7, 4, 8, 1, 9, 1, 10, 5, 11, 1, 12, 2, 13, 7, 14, 1, 15, 1, 16, 8, 17, 3, 18, 1, 19, 10, 20, 1, 21, 1, 22, 11, 23, 1, 24, 2, 25, 13, 26, 1, 27, 6, 28, 14, 29, 1, 30, 1, 31, 16, 32, 7, 33, 1, 34, 17, 35, 1, 36, 1, 37, 19

Offset

1,4

Comments

See A279119 for the same sequence with numbers including 0.

See A055396 for a similar sequence where all equal terms share a factor > 1.

Links

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Formula

a(n) = 1 + A279119(n). - Rémy Sigrist, Mar 04 2024

Example

a(4)=2 because if we had a(4)=1, then i=2 and i=4, which are not coprime indices, would have the same value 1. So a(4)=2, which is a first occurrence.
a(9)=2 because if we had a(9)=1, i=3 and i=9, would have the same value despite not being coprime indices. a(9) can be 2 because the only other index with a 2 is a(4)=2 and 4 is coprime to 9.
a(15)=4 because 4 is the smallest value such that every previous index at which a 4 occurs is coprime to i=15. In this case, 4 has only occurred at i=8 and 8 is coprime to 15.

Prog

(Python)
from math import gcd, lcm
from itertools import combinations as C, count, islice
def agen(): # generator of terms
    yield from [1, 1, 1]
    lcms = {1: 6}
    for n in count(4):
        an = next(an for an in count(1) if an not in lcms or gcd(lcms[an], n) == 1)
        yield an
        if an not in lcms: lcms[an] = n
        else: lcms[an] = lcm(lcms[an], n)
print(list(islice(agen(), 75))) # Michael S. Branicky, Mar 02 2024

Crossrefs

Cf. A100480, A279119, A370408.

Sequence in context: A246277 A260739 A130747 * A055440 A250028 A101279

Adjacent sequences: A370819 A370820 A370821 * A370823 A370824 A370825

Keyword

nonn

Author

Neal Gersh Tolunsky, Mar 02 2024

Extensions

a(22) and beyond from Michael S. Branicky, Mar 02 2024

Status

approved